Cantor’s Theorem: The Movie

Ed Talavera is the chair of the Department of Cinema and Interactive Communication in the University of Miami School of Communication. He is an award-winning cinematographer and director of narrative and documentary feature films. His work has aired on HBO, Showtime, Cinemax, and theaters worldwide. He was the cinematographer for a wonderful film about food that aired on DirecTV, “Mistura.” The film is about how a food festival unites the diverse nation of Peru, but what we notice in the trailer is the food. Gorgeous food.

Today Ken and I wish to announce a new film directed by Ed on Cantor’s Theorem.

We were the scriptwriters, Ed was the producer and director, and his team of students were the actors and crew. Our film is a 7-minute “short,” and you can skip our comments and go directly to see it via YouTube right here or Vimeo at links below. You won’t see it at the Cannes Film Festival, but perhaps it will make the Canes Film Festival, which Ed directs.

It’s Not A Course Video

There is a great deal of interest in video-based courses these days. We have already talked about them here and here, and a Google search will turn up many links related to on-line education.

The usual video-based course is a film of an instructor talking and writing at a board or on a tablet computer. These courses are popular among students, at least partially because they are free. The videos are informative, although it is yet unclear whether they are as good or better than physical courses. You know—course with students in seats and an instructor talking and interacting with them. We will see.

The GLL philosophy is always to do things in a novel manner. After all both of us believe that factoring is easy, and one believes that is possible. Our approach to video based courses has been to be different, not just to film a lecture, but to do something else. Indeed to express this philosophy, several of our draft scripts begin with the end of a lecture. It is possible that this will become an “invariant” like several we have for style on this blog.

“Factors”

Our plan is to make a mini-series length course with characters who have issues, who follow an interesting story line—vampires?—and yet are able to convey the information we want the students to learn. Our view is to create a new type of film: not a documentary, not a docudrama, not a dry lecture. A mixture of fiction and information.

In order to make this happen Ken and I have started to write movie scripts. We are not experts at writing anything, and are rank novices at writing scripts. At GLL we never let something like this stop us from trying, so we have been busy writing a pilot script for our series. The script is available for those who might want to help us see if it is on track.

If you wish us to state the series’ theme in 25 words or fewer, we might try:

What important parts of our lives depend on what we know about numbers with hundreds of digits, and other parts of theory?

We want to get people excited about the concepts, and to know about “factors” affecting how the applications operate and when they are safe. These include secure commerce, online social systems, handheld and sensing devices, and certain numerical applications.

The test of whether we are on to something cool or wasting our time is best determined by seeing at least a pilot film. We have tried to get help—we may be able to hack out a plausible script—but we have no idea how to make a film. Enter Ed. Dick met Ed last year and explained the basic idea to him. He immediately was excited and said:

Let’s make a movie.

He did.

The Movie

Our pilot script for the mini-series itself was too long, and had many actors and many scenes. So following Ed’s suggestion we wrote a much shorter script about Cantor’s Theorem. This is the script that Ed filmed this past December in Miami.

Ken and I were both there, and were thrilled to see the production aspects of film-making. It is a very cool experience. Three Miami students acted, and three others formed the crew. The scenes were filmed in reverse order of occurrence in the script, so we started at 8am in one of the university area’s best-known bars. We gave the actors space and left all details to Ed’s direction, but we interacted a fair bit with the crew, including being fascinated by the second-by-second positioning of a large arm-held microphone to obtain accurate sound. It seems statistically impossible that you don’t see this microphone in any of the frames, but Ed knows his camera.

The result is the film that is now on Vimeo and YouTube. We hope you like it. Full credits to the actors and crew members, whom we thank for their eye-opening efforts, are listed at the Vimeo links (click “see all”): mobile, SD, and HD.

Open Problems

The main open problem is: should we continue to go in this direction? Should we continue going in this direction of fiction and instructional films? What do you think? Is it two thumbs up? Or not?

One last point is, what do a director and a producer do? Maggie Smith in Gosford Park says:

They’re rather a mixed bunch. That Mr. Weissman’s very odd. Apparently, he produces motion pictures. The Charlie Chan Mysteries. Or does he direct them? I never know the difference.

Our apologies to Ed, since I think we now know the difference. Thanks to Ed for doing both jobs so well.

I am not entirely sure… but I think both authors of the blog are sitting near the bar. Ken faces the camera, Dick is not. Also by the way they look down and I think some arm movement, it looks as if they are playing chess?

By the way… I was expecting the part 0.999999999999999.. = 1.000000000000000000000.. to come up, but no 😛

The idea is awesome, and the execution is just fine, but the math itself has turned into what Alfred Hitchcock called “the McGuffin”: the thing that is the focus of activity, but is not really what the movie is all about. Making the movie /about/ student-student interactions and the need to know something “for the test” isn’t going to hold up for an entire series. As education moves from classrooms full of college kids to online resources for any learner, we’d wish for the story line to be mathematics proper. Aren’t real mathematicians motivated by the need to discover and prove true things? You’ve given two characters a yen for clarifying their explanations, and they then model skills of mathematical thought, but why are they thus? Can the series come to be about what makes thought clear, and the powerful way definiteness and certainty gives mathematics its subjective value to humans?

I found the movie charming, but I think you are right, the fact that the math is a topic of conversation for the characters is less immediately compelling than the human relationships. These students’ goals are to pass their classes and have fun with their friends, the problem they are working on is incidental to that.

Dramatizations of say, historical or science-fictional scenarios where the characters face a problem that can only be fixed by their comprehension and proof of some theorem would center the drama around the math instead. I am thinking of many fascinating stories I have read from the first two World Wars involving logistics, ballistics, encryption, etc. Just for example.

Among plot devices and focal subjects we are considering are:
1. Undergraduate makes major breakthrough on factoring by solving a mis-stated summer practice homework problem.
2. Undergraduate has to cope with identity being spoofed (or stolen) on forums,
3. After giving a lecture showing how to use Euclid’s Algorithm to find a common factor of two numbers if it exists, a professor gets a new idea related to the Miller-Rabin algorithm and Lenstra’s attack.
4. Twists like in the movie Travelling Salesman, but perhaps conjectural not actual, and private on-campus rather than in a government research center.
5. How the idea of complexity of numbers might relate to 3. and to the ACC problem for the Mo”bius function (see this).

Little things include illustrating the difference between linear, quadratic, and cubic running times, set up by noting that the US national debt equals a stack of pennies all the way out to Uranus, but if you throw the pennies in a pit instead, it would only need to be 3/5 of a mile on the side. Hence a decent-size zinc mine—or a smaller copper mine—could pay off our debt!

The following is a slightly lengthier version that (basically) captures P \ne NP.

A proof based on the presented problem from anybody is welcome. However, one based on something similar or even totally different would be better. Please be sure to make the proof totally computational model independent covering, among other things, QC and DNAC.

Thanks

=== Problem for the Proper Containment ===

Suppose there is an electric source and a lamp post some distance away.

You are given n (say 300) pieces of wire that, when all connected, can just reach the lamp post and light the lamp.

There are also n pieces of wire that are broken inside, yet from the outside the wires, good and bad, all look indistinguishable.

If the good and bad wires are mixed up and you try to connect up only the good ones to light up the lamp, the problem you face is similar to the separation problem posed in Language[1], only that the latter would need more effort as order is relevant.

If you connect any bad wire, the electric current cannot go through, but you would not know which individual one is good or bad.

—
[1] Language refers to a more formal version of the problem that will be available when everything is made public in a short while.

=== End of Problem ===

P.S.

If the mere problem description do not give you enough to proceed and if you agree to viewing additional details only privately till I get ready to make everything public, please contact tcne1837@hotmail.com with the subject title (without quotes) “request for details for own consumption”. Thanks for your interest.

===========

Dick,

Did not plan the jump to P \ne NP directly.

There must be either MITMA (man-in-the-middle attack) or ‘Never got’ is somehow not the entire truth. Did you every get the questions, though?

I like your movie very much. However I wouldn’t say that it’s not a course video. The actors are really giving us a fine lecture about Cantor’s proof in a quite pleasant way.

> “After all both of us believe that factoring is easy, and one believes that P=NP is possible.”

Factoring might be easy, but finding the corresponding algorithm seems to be almost impossible in the sense of probability theory.

Just as well, quantum computing might be possible but not feasible. The notion of a working quantum computer might not break any law but the probability that living beings will ever build one is null. By the way, there might already be full-fledged quantum computers in Nature. Whether the human brain is one of them remains – as far as I know – an open question.

Likewise, natural intelligence is possible too. We can notice intelligent people among us. Yet, building an intelligent mind from scratch doesn’t seem to be feasible either.

I think the conventional wisdom about this kind of problems has long been misguided. The main issue shouldn’t be whether something can exist, it should be whether we – as living creatures which are the product of millions of years of evolution – can realize these things from scratch. If I were religious-minded I’d be tempted to conjecture that “creatures are always inferior to their creators”. But this also applies to the case where the creator is a computer, no matter whether it be natural or artificial.

The basic idea, of having someone who shares the possible confusions of the viewer and articulates them in the video, is one that I strongly agree with. It is a much better approach than a video that just lectures at you. I would be much more excited about things like the Khan academy if they based their videos on this principle.

It’s true that the benefits of using movies for teaching math have been much underrated until now. As we can see, the full power of this method requires good actors such as the three of these. So quite likely, the future of math teaching lies in video. After all, teachers have always had to be good actors…

I like movie, I like the idea. Are you planning to start polyMovieth? So, that math people can write scripts, and movie people can shoot it? Funny and useful open source activity.

By the way, Cantor only show that there are no sequence of the digits in the list, but it does not show, there are no such number in the list with different representation. It is like everybody just forget about Cauchy.

I really like that you were involved in making this video. I can’t say that I love it, there are things I like and there are things that I would constructively criticize. But, I love the idea and look forward to what’s coming next. I myself am a beginner photographer and enjoy seeing Mathematicians or Computer Scientists that diversify in their interests and projects.

I found that quite painful, to be honest. I don’t think that couching a lecture in the form of a dialog (a la Socrates) will pass the believability test for a broader audience. And the bookend lesson that attention to math will lose you a chance at sex is a crippling kick in the nuts.

Is there any reason to believe P!=NP is possible? Most folks would say a proof is a proof only if the proof is formally accepted as a proof. A proof is found only if it is verified as a proof. Isnt it? Philosophically why should a certificate be different from a proof?

>>> Is there any reason to believe P!=NP is possible?
It is not a question whether there is reason. It was proven quite a while back.

>>> Most folks would say a proof is a proof only if the proof is formally accepted as a proof. Isnt it?
Yes and no. If the idea is right, it does not have to be formal. Therefore, the lemma-lemma-theorem (or theolem if one prefers) is a ritual and has little to do with the nature of truth. However, if the idea is right, it must be able to be formalized. So, yes. It has to be ‘formally’ (not in the sense of a ritual) accepted. Jokingly, if you do not do it formally, you may leave imaginary holes which will happily find happy opportunistic pluckers. 🙂 Not guaranteed even if you do it formally. 😦

>>> Philosophically why should a certificate be different from a proof?
What do you exactly mean by a certificate? An existential proof? Characteristics of a valid proof? ‘Enough’ evidence?

>>> A proof is found only if it is verified as a proof. Isnt it?
NO. A proof is a proof, regardless if it is verified. If the world refuses to verify a proof, it won’t change it being a proof. If Sir Eddington had not traveled to Africa, we wouldn’t have Relativity? I doubt. Truth is independent of human activities.

Mathematicians refuse to look at the real problem of Cantor’s proof. Why would you think this video would be any different? It doesn’t help people who don’t believe in Cantor’s “proof” because none of the things mentioned is what people have problems with.

Most people who disagree with Cantor believe that the digits are a proper subset of the enumeration no matter if it’s a finite list, N or R (unless using the infinite identity matrix, but that applies to N only). In Cantor’s proof, they completely bypass that part of the proof. In fact, we know that with N, there is no bijection between the digits and the rows. Cantor just proved that the same is true of R. It’s an expected result. This is why people have a problem with Cantor’s proof. Cantor is showing an EXPECTED result. The interpretation is what’s different. In the video, they claim Cantor showed that the reals are uncountable. Others (uncounts) say that he showed that there is a surjection from the digits to the rows of any infinite list (specifically R in this case, but that it’s true of N as well).

Remember, bijections are a possibility, not a requirement between two sets. The digits define a mapping to the rows that is not a bijection for all infinite sets (except for those listed in the form of an infinite identity matrix). Use a different mapping and you can get a bijection.

Also, the who goes first thing is how you tell if someone is using circular logic. Another strike against Cantor.

—–
I will simplify the issue even FURTHER. 1 digit, 2 rows. 2 digits, 4 rows. 3 digits, 8 rows. Surjection from digits to rows. Yes, it is a finite list. So what? Even if I continue to infinity, the first three digits will still map to 8 rows. That’s not going to change just because we have an infinite list. In fact, Cantor proved this surjection is true with R as well. And it’s trivial that this is true of N though mathematicians still refuse to even look at it. There you go. Cantor proved a fact of R that is true in the finite world and with N. But for some inexplicable reason, we’re supposed to disregard that and give it a meaning COMPLETELY unrelated to what we already know.

That’s the problem. Cantor’s conclusion is completely unrelated to what he showed. It’s taken out of thin air.

Ok, so I like the video, but I can’t help thinking that you can’t make a complete list of integers either. You could always add 1 to the biggest number in the list, and then you’ve got a new number that wasn’t on your list.
Is this just me being stupid?

You’re not being stupid – it’s just that the lists they’re talking about are supposed to be infinite. The purpose of their proof is to show that even infinite lists aren’t long enough to count the reals. Actually it’s logically possible to “count” them with the uncountable ordinals, but they make no mention of this fact.